We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Can you make a 3x3 cube with these shapes made from small cubes?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

What is the best way to shunt these carriages so that each train can continue its journey?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

What happens when you try and fit the triomino pieces into these two grids?

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Exchange the positions of the two sets of counters in the least possible number of moves

How many different triangles can you make on a circular pegboard that has nine pegs?

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Move just three of the circles so that the triangle faces in the opposite direction.

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?